Let $X$ be a Riemannian manifold. If $X$ is simply connected, irreducible, and not a symmetric space then we know that the possible holonomy groups of the metric on $X$ are:

1) $O(n)$ General Riemannian manifolds

2) $SO(n)$ Orientable manifolds

3) $U(n)$ Kahler manifolds

4) $Sp(n)Sp(1)$ Quaternionic Kahler manifolds

5) $SU(n)$ Calabi-Yau manifolds

6) $Sp(n)$ Hyperkahler manifolds

7) $G_{2}$ (if $X$ has dimension 7) $G_{2}$ manifolds

8) $Spin(7)$ (if X has dimension 8) $Spin(7)$ manifolds

Of these, cases 5-8 are important in string theory. The reduced holonomy implies that these manifolds have vanishing Ricci curvature and hence are automatically solutions to Einstein's equations of general relativity.

However, physics does not take place on a Riemannian manifold, but rather on a Lorentzian one. Thus my question is: what is known about special holonomy manifolds for metrics of general signature? (I am most interested in the (1,n) case!) In particular I would like to know if there is a classification of allowed holonomy groups and further if there are interesting examples that I should be aware of.

2 Answers
2

I think it would be more accurate to say that the real reason why
Calabi-Yau, hyperkähler, $G_2$ and $\mathrm{Spin}(7)$ manifolds are of
interest in string theory is not their Ricci-flatness, but the fact
that they admit parallel spinor fields. Of course, in
positive-definite signature, existence of parallel spinor fields
implies Ricci-flatness, but the converse is still open for compact
riemannian manifolds, as discussed in this
question, and known to fail for noncompact manifolds as pointed
out in an answer to that question.

The similar question for lorentzian manifolds has a bit of history.
First of all, the holonomy principle states that a spin manifold
admits parallel spinor fields if and only if (the spin lift of) its
holonomy group is contained in the stabilizer subgroup of a nonzero
spinor. Some low-dimensional (i.e., $\leq 11$, the cases relevant to
string and M-theories) investigations (by Robert Bryant and myself,
independently) suggested that these subgroups are either of two types:
subgroups $G < \mathrm{Spin}(n) < \mathrm{Spin}(1,n)$, whence $G$ is
the ones corresponding to the cases 5-8 in the question, or else $G =
H \ltimes \mathbb{R}^{n-1}$, where $H < \mathrm{Spin}(n-1)$ is one of
the groups in cases 5-8 in the question. Thomas Leistner showed that
this persisted in the general case and, as Igor pointed out in his
answer, arrived at a classification of possible lorentzian holonomy
groups. Anton Galaev then constructed metrics with all the possible
holonomy groups, showing that they all arise. Their work is reviewed
in their
paper (MR2436228).

The basic difficulty in the indefinite-signature case is that the de
Rham decomposition theorem is modified. Recall that the de Rham
decomposition theorem states that if $(M,g)$ is a complete, connected
and simply connected positive-definite riemannian manifold and if the
holonomy group acts reducibly, then the manifold is a riemannian
product, whence it is enough to restrict to irreducible holonomy
representations. This is by no means a trivial problem, but is
tractable.

In contrast, in the indefinite signature situation, there is a
modification of this theorem due to Wu, which says that it is not
enough for the holonomy representation to be reducible, it has to be
nondegenerately reducible. This means that it is fully reducible
and the direct sums in the decomposition are orthogonal with respect
to the metric. This means that it is therefore not enough to restrict
oneself to irreducible holonomy representations. For example,
Bérard-Bergery and Ikemakhen proved that the only lorentzian holonomy
group acting irreducibly is $\mathrm{SO}_0(1,n)$ itself: namely, the
generic holonomy group.

It should be pointed out that in indefinite signature, the
integrability condition for the existence of parallel spinor fields is
not Ricci-flatness. Instead, it's that the image of the Ricci
operator $S: TM \to TM$, defined by $g(S(X),Y) = r(X,Y)$, with $r$ the
Ricci curvature, be isotropic. Hence if one is interested in
supersymmetric solutions of supergravity theories (without fluxes) one
is interested in Ricci-flat lorentzian manifolds (of the relevant
dimension) admitting parallel spinor fields. It is now not enough to
reduce the holonomy to the isotropy of a spinor, but the
Ricci-flatness equation must be imposed additionally.

I am no expert in Lorentzian geometry, but according to MathSciNet review, Lorentzian holonomy groups has been completely classified by Thomas Leistner in [On the classification of Lorentzian holonomy groups. J. Diff. Geom. 76 (2007), no. 3, 423--484] who builds on classical work of Berger.